If by "BT3 modem router" you mean a BT Home Hub 3, I have found that it at times refuses to support a wired Ethernet connected device (my pc for example which is normally connected by a Ethernet cable direct to the hub) and a wireless connected streaming device at the same time. Other wireless devices will also show poor performance once the streaming device is connected. Changing to a wired connection once in this state does not improve matters. In time it might but I have never given it the chance.
In this condition the home hub dashboard will show connected at your normal sync rate to the exchange but a speed test will give a slow speed result and some devices may fail to show any speed test result or even load the page.
The quick way to get things working as normal is to reset the home hub.
Your situation may be different but the above describes my situation and experience with a Home Hub 3.